Asymptotics for credit portfolio losses due to defaults in a multi-sector model

Abstract

Consider a credit portfolio with the investments in various sectors and exposed to an external stochastic environment. The portfolio loss due to defaults is of critical importance for social and economic security particularly in times of financial distress. We argue that the dependences among obligors within sectors (intradependence) and across sectors (interdependence) may coexist and influence the portfolio loss. To quantify the portfolio loss, we develop a multi-sector structural model in which a multivariate regular variation structure is employed to model the intradependence within sectors, and the interdependence across sectors is implied in the arbitrarily dependent macroeconomic factors, although, given them, obligors in different sectors are conditionally independent. We establish some sharp asymptotic formulas for the tail probability and the (tail) distortion risk measures of the portfolio loss. Our results show that the portfolio loss is mainly driven by the latent variables and the recovery rate function, and is also potentially affected by the macroeconomic factors and the intradependence within sectors. Moreover, we implement intensive numerical studies to examine the accuracy of the obtained approximations and conduct some sensitivity analysis.

Appendix A: Proofs

For convenience, we abbreviate the conditional probability measure $P(\cdot |{\varvec{S}})$ to $P^{\varvec{S}}(\cdot )$, and for each $i=1,\dots , k$, construct an associated random vector $\varvec{Y}_i=(Y_{i 1},\ldots ,Y_{i d_i})$ with

$$\begin{aligned} Y_{i j} = \frac{1}{R\left( \frac{X_{i j}}{a_{i j}} -1 \right) }, \end{aligned}$$

(A.1)

$j=1,\dots , d_i$. The following lemma can be found in Lemma 4.3 of Tang and Yuan (2013), which is crucial in the proof of Theorem 3.1.

Lemma A.1

If Assumption 3.1(ii) is satisfied and $R\in \textrm{RV}_{-\beta }$ for some $0<\beta \le \infty$, then it holds that for each $i=1,\dots , k$, almost surely,

$$\begin{aligned} \frac{P^{\varvec{S}}\left( \varvec{Y}_{i}/x \in \cdot \right) }{\overline{F_i}\circ \left( {1}/{R} \right) ^{\leftarrow }(x)} {\mathop {\rightarrow }\limits ^{v}} \tilde{\nu }^{\varvec{S}}_{i} (\cdot ) \end{aligned}$$

on $[0,\infty ]^{d_i}\setminus \{\varvec{0}\}$, where $\tilde{\nu }^{\varvec{S}}_{i}$ is defined by (3.3).

Proof of Theorem 3.1

We start with a claim that, almost surely,

$$\begin{aligned} \lim _{x\rightarrow \infty }\frac{P^{\varvec{S}}\left( L>1-1/x \right) }{\left( \prod _{i=1}^{k}\overline{F_i} \right) \circ \left( 1/R \right) ^{\leftarrow } (x)} =\frac{\prod _{i=1}^{k}\Gamma \left( 1+\frac{\alpha _i}{\beta } \right) }{\Gamma \left( 1+\frac{1}{\beta }\sum _{i=1}^{k}\alpha _i\right) } \prod _{i=1}^k \tilde{\nu }^{\varvec{S}}_{i}(A_{i}). \end{aligned}$$

(A.2)

Recalling $\varvec{Y}_{i}, i=1,\dots , k$, in (A.1) and $\sum _{i=1}^k\sum _{j=1}^{d_i} e_{i j} =1$, we rewrite the conditional probability $P^{\varvec{S}}\left( L>1-\frac{1}{x} \right)$ as

$$\begin{aligned} P^{\varvec{S}}\left( L>1-\frac{1}{x} \right) =P^{\varvec{S}}\left( \sum _{i=1}^k \sum _{j=1}^{d_i} \frac{e_{i j}}{ Y_{i j}/x} <1 \right) . \end{aligned}$$

If we can prove that, almost surely, for any $e_{i j}>0$, $j=1,\dots , d_i$; $i=1,\dots , k$,

$$\begin{aligned} P^{\varvec{S}}\left( \sum _{i=1}^k \sum _{j=1}^{d_i} \frac{e_{i j}}{ Y_{i j}/x} <1 \right) \sim \frac{\prod _{i=1}^{k}\Gamma \left( 1+\frac{\alpha _i}{\beta } \right) }{\Gamma \left( 1+\frac{1}{\beta }\sum _{i=1}^{k}\alpha _i\right) } \prod _{i=1}^k \tilde{\nu }^{\varvec{S}}_{i}(A_{i})\cdot \left( \prod _{i=1}^{k}\overline{F_i} \right) \circ \left( \frac{1}{R} \right) ^{\leftarrow } (x), \end{aligned}$$

(A.3)

then (A.2) holds. For (A.3), we proceed by induction on k. Trivially, (A.3) holds for $k=1$ from Lemma A.1. Now we assume by induction that it holds for all $k\le k_0$ and some $k_0\in \mathbb N$, and we are going to prove it for $k=k_0 +1$. First consider the lower bound of (A.3) with $k=k_0 +1$. For each $i=1,\dots , k_0+ 1$, by (3.4) it is easy to see that ${\varvec{y}}_1 \in A_i$ implies ${\varvec{y}}_2 \in A_i$ for any ${\varvec{y}}_2 \ge {\varvec{y}}_1$, and thus $t (\partial A_i)\cap \partial A_i = \emptyset$ for every $t>1$. Then, by Lemma 7.1 of Shi et al. (2017), we have $\tilde{\nu }^{\varvec{S}}_i( \partial A_i )=0$. For any large but fixed $n\in \mathbb N$, almost surely,

$$\begin{aligned}{} & {} P^{\varvec{S}}\left( \sum _{i=1}^{k_0+1} \sum _{j=1}^{d_i} \frac{e_{ij}}{Y_{i j}/x}<1 \right) \nonumber \\\ge & {} \sum _{m=1}^{n-1} P^{\varvec{S}}\left( \frac{m-1}{n} \le \sum _{i=1}^{k_0} \sum _{j=1}^{d_i} \frac{e_{ij}}{ Y_{ij}/x}< \frac{m}{n}, \sum _{j=1}^{d_{k_0+1}} \frac{e_{(k_0+1) j}}{ Y_{(k_0+1) j}/x}<1-\frac{m}{n} \right) \nonumber \\= & {} \sum _{m=1}^{n-1} \left( P^{\varvec{S}}\left( \sum _{i=1}^{k_0}\sum _{j=1}^{d_i} \frac{e_{i j}}{ Y_{i j}/x}< \frac{m}{n}\right) - P^{\varvec{S}}\left( \sum _{i=1}^{k_0}\sum _{j=1}^{d_{i}} \frac{e_{i j}}{ Y_{i j}/x}< \frac{m-1}{n}\right) \right) \nonumber \\{} & {} \times P^{\varvec{S}} \left( \sum _{j=1}^{d_{k_0 +1}} \frac{e_{(k_0+1) j}}{ Y_{(k_0+1) j}/x} <1-\frac{m}{n} \right) \nonumber \\\sim & {} \left( \sum _{m=1}^{n-1} \left( 1-\frac{m}{n}\right) ^{\frac{\alpha _{k_0+1}}{\beta }} \left( \left( \frac{m}{n}\right) ^{\frac{1}{\beta }\sum _{i=1}^{k_0}\alpha _i} -\left( \frac{m-1}{n}\right) ^{\frac{1}{\beta }\sum _{i=1}^{k_0}\alpha _i} \right) \right) \cdot \frac{ \prod _{i=1}^{k_0}\Gamma \left( 1+ \frac{\alpha _i}{\beta }\right) }{\Gamma \left( 1+\frac{1}{\beta } \sum _{i=1}^{k_0}\alpha _i\right) }\nonumber \\{} & {} \times \prod _{i=1}^{k_0+1}\tilde{\nu }^{\varvec{S}}_{i}(A_{i}) \left( \prod _{i=1}^{k_0+1}\overline{F_i} \right) \circ \left( \frac{1}{R} \right) ^{\leftarrow } (x) \end{aligned}$$

(A.4)

where we used (A.3) for all $k\le k_0$ and (2.4) in the last step. Letting $n\rightarrow \infty$, the sum in (A.4) converges to $\int _0^1 (1-y)^{\frac{\alpha _{k_0+1}}{\beta }} \textrm{d} \left( y^{\frac{1}{\beta }\sum _{i=1}^{k_0}\alpha _i}\right) ={\Gamma \left( 1+\frac{1}{\beta }\sum _{i=1}^{k_0}\alpha _i\right) \Gamma \left( 1+ \frac{\alpha _{k_0 +1}}{\beta }\right) }/{\Gamma \left( 1+\frac{1}{\beta } \sum _{i=1}^{k_0 +1}\alpha _i\right) }$. For the upper bound of (A.3) with $k=k_0+1$, we do the split

$$\begin{aligned}{} & {} P^{\varvec{S}}\left( L>1-\frac{1}{x} \right) \\{} & {} \quad \le \sum _{m=1}^{n} P^{\varvec{S}}\left( \frac{m-1}{n} \le \sum _{i=1}^{k_0}\sum _{j=1}^{d_i} \frac{e_{i j}}{ Y_{i j}/x}< \frac{m}{n}, \sum _{j=1}^{d_{k_0 +1}} \frac{e_{(k_0+1) j}}{ Y_{(k_0+1) j}/x} <1-\frac{m-1}{n} \right) . \end{aligned}$$

Then, a similar discussion to (A.4) can be carried out to derive the upper bound. Thus, the desired relation (A.3) holds for $k=k_0+1$.

Therefore, applying the dominated convergence theorem and by (A.2) we can obtain

$$\begin{aligned} \lim _{x\rightarrow \infty }\frac{P\left( L>1-1/x \right) }{\left( \prod _{i=1}^k \overline{F_i} \right) \circ \left( 1/R \right) ^{\leftarrow } (x)}= & {} \lim _{x\rightarrow \infty }E\left[ \frac{P^{\varvec{S}}\left( L>1-1/x \right) }{\left( \prod _{i=1}^k \overline{F_i} \right) \circ \left( 1/R \right) ^{\leftarrow } (x)} \right] \nonumber \\= & {} \frac{\prod _{i=1}^k\Gamma \left( 1+\frac{\alpha _i}{\beta }\right) }{\Gamma \left( 1+\frac{1}{\beta }\sum _{i=1}^k\alpha _i\right) }E\left[ \prod _{i=1}^k \tilde{\nu }^{\varvec{S}}_{i}(A_{i}) \right] . \end{aligned}$$

In the derivation above, the applicability of the dominated convergence theorem can be justified as follows. For any fixed $x\ge 1$, denote by

$$\Delta _{i}(x) = \left\{ \varvec{x}\in [\varvec{0}, \varvec{\infty }]^{d_i}: \sum _{j=1}^{d_i} e_{i j} R \left( \frac{x_j}{a_{i j}} -1 \right) <\frac{1}{x} \right\} ,$$

$i=1,\dots , k$. Clearly, it can be verified that, for any fixed $x\ge 1$, the sets $\Delta _i(x), i=1,\dots , k$, are all increasing, since $R(\cdot )$ is a non-increasing function. By Assumption 3.1(i), (iii), for any fixed $\epsilon >0$, there exists a large constant $M>0$, such that for all sufficiently large $x\ge 1$, almost surely,

$$\begin{aligned} \frac{P^{\varvec{S}}\left( L>1-1/x \right) }{\left( \prod _{i=1}^k \overline{F_i} \right) \circ \left( 1/R \right) ^{\leftarrow } (x)}\le & {} \frac{\prod _{i=1}^k P^{\varvec{S}}({\varvec{X}}_i \in \Delta _i(x))}{\left( \prod _{i=1}^k\overline{F_i} \right) \circ \left( 1/R \right) ^{\leftarrow } (x)} \nonumber \\\le & {} \frac{\prod _{i=1}^k P({\varvec{X}}_i^* \in \Delta _i(x))}{\left( \prod _{i=1}^k \overline{F_i} \right) \circ \left( 1/R \right) ^{\leftarrow } (x)}\nonumber \\\le & {} \frac{\prod _{i=1}^k P\left( e_{i 1} R \left( \frac{X^*_{i 1}}{a_{i 1}} -1 \right) <\frac{1}{x} \right) }{\left( \prod _{i=1}^k \overline{F_i} \right) \circ \left( 1/R \right) ^{\leftarrow } (x)}\nonumber \\\le & {} \prod _{i=1}^k \frac{ P\left( X^*_{i 1} > (1-\epsilon ) e_{i 1}^{\frac{1}{\beta }} a_{i 1} \left( 1/R \right) ^{\leftarrow } (x) \right) }{ \overline{F_i} \circ \left( 1/R \right) ^{\leftarrow } (x) }\nonumber \\\le & {} 2\prod _{i=1}^k \nu ^{*}_i((\varvec{1}_1, \varvec{\infty }]) \max \left\{ 1, M\big ( (1-\epsilon ) e_{i 1}^{\frac{1}{\beta }} a_{i 1} \big )^{-(\alpha _i+ \epsilon )}\right\} ,\nonumber \\ \end{aligned}$$

(A.5)

which is integrable with respect to $P(\varvec{S}\in \textrm{d}\varvec{s})$. Here we used (2.5) in the fourth step, and (2.2) in the last step. This completes the proof of Theorem 3.1.$\square$

Proof of Theorem 3.2

Combined with Corollary 3.1, for relation (3.6), it suffices to prove that

$$\begin{aligned} 1-\textrm{D}_{g_q} (L)\sim \int _{[0,1]} u^{\frac{\beta }{\sum _{i=1}^k \alpha _i}} g(\textrm{d}u)\cdot \left( 1-\textrm{VaR}_q(L) \right) . \end{aligned}$$

(A.6)

Note that for any $0<q<1$,

$$\begin{aligned} 1-\textrm{VaR}_q(L)=1-F_L^\leftarrow (q)=\frac{1}{\left( 1/\overline{F_{\frac{1}{1-L}}}\right) ^\leftarrow \left( \frac{1}{1-q}\right) }. \end{aligned}$$

Since the distortion function g is left continuous, so is $g_q$. Then, by Theorem 6 of Dhaene et al. (2012), we have

$$\begin{aligned} \frac{1-D_{g_q}(L)}{1-\textrm{VaR}_q(L)}= & {} \int _{[0,1]} \frac{1-F_L^{\leftarrow } (1-(1-q)u)}{1-F_{L}^{\leftarrow } (q)} g(\textrm{d}u)\nonumber \\= & {} \int _{(0,1]} \frac{ \left( {1}/{\overline{F_{\frac{1}{1-L}}}} \right) ^{\leftarrow } \left( \frac{1}{1-q} \right) }{ \left( {1}/{\overline{F_{\frac{1}{1-L}}}} \right) ^{\leftarrow } \left( \frac{1}{(1-q)u} \right) } g(\textrm{d}u). \end{aligned}$$

(A.7)

We first consider the convergence of the integrand in (A.7) as $q\uparrow 1$. By $\overline{F_i}\in \textrm{RV}_{-\alpha _i}, i=1,\dots ,k$, $R\in \textrm{RV}_{-\beta }$, and $E \left[ \prod _{i=1}^k \tilde{\nu }^S_{i}(A_{i})\right] >0$, Theorem 3.1 implies that $1/\overline{F_{\frac{1}{1-L}}}\in \textrm{RV}_{\frac{1}{\beta }\sum _{i=1}^k\alpha _i}$. Then, applying Proposition 0.8(v) of Resnick (1987) gives that for any fixed $u\in (0,1]$ and $0\le \frac{1}{\beta }\sum _{i=1}^k\alpha _i\le \infty$,

$$\begin{aligned} \lim _{q \uparrow 1} \frac{ \left( {1}/{\overline{F_{\frac{1}{1-L}}}} \right) ^{\leftarrow } \left( \frac{1}{1-q} \right) }{ \left( {1}/{\overline{F_{\frac{1}{1-L}}}} \right) ^{\leftarrow } \left( \frac{1}{(1-q)u} \right) }= u^{\frac{\beta }{ \sum _{i=1}^k \alpha _i }}. \end{aligned}$$

Therefore, the desired relation (A.6) can be derived by applying the dominated convergence theorem. Indeed, for all $u\in (0,1]$ and $q\in (0,1)$,

$$\begin{aligned} \frac{ \left( {1}/{\overline{F_{\frac{1}{1-L}}}} \right) ^{\leftarrow } \left( \frac{1}{1-q} \right) }{ \left( {1}/{\overline{F_{\frac{1}{1-L}}}} \right) ^{\leftarrow } \left( \frac{1}{(1-q)u} \right) }\le 1, \end{aligned}$$

which is integrable with respect to $g(\textrm{d}u)$ in $u\in [0,1]$. It ends the proof of Theorem 3.2.$\Box$